This is the same as sim 3, but with a 20 degree relative inclination (star and companion in same plane, planet inclined at 20░). Just to see what happens if the inclination is between zero and the critical value of 39 degrees.

And it's weird. (I'm doing this at a 2048 timestep, so hopefully that's not making it look so different - it didn't seem to have a noticeable effect on sim 4)

ok , thats fine , I can calculate the positions with this , however calculating the initial velocities gives some more problems while two big masses are involved and thus the sun is not fixed ...Any chance to get the excel-table with the initial velocities ?

ok , thats fine , I can calculate the positions with this , however calculating the initial velocities gives some more problems while two big masses are involved and thus the sun is not fixed ...Any chance to get the excel-table with the initial velocities ?

Gravity Simulator calculates the initial velocities for you, based on the orbital elements. Start the simulation paused, then go to Object > Edit Object, and the position and velocity vectors of each object will be visible.

... Long time ago that we met Kozai ... Is this following system a kind of Kozai influence ? If its not it shows a lot of dynamics , analog to Kozai.

System consists of 2 "our" suns , orbiting each other at 0.2 AU , with a small eccentricity . An Earth sized planet is put in polar orbit around them at 1AU , also with a small eccentricity . The Earth planet leaves its polar orbit and regains it after about 350 years. Screen shots were made every 10 years . In the first part the system is viewed side-on, in the second part from above .

Tony gave a formula to calculate the period of the Kozai cycle in one of the other posts . It looks like áPe= .....(1-e2^2) , among also factors with masses and major axis . It looks as if bodies with high eccentricities have small periods . So such bodies should show remarkable perturbations on small time scales . I selected the asteroid 2006HY51 which has the biggest known eccentricity of all asteroids . Running the sim with only Jupiter and Saturn as perturbers I get the following picture .... It is clear that the asteroid 2006HY51 has a very short period ( near 2000 years) (Note : i and e are both multiplied by 10E4 here )

Wondering what the influence is of this rather quickly changing eccentricity and inclination on the threat to earth for collisions I ran the sim of 2006HY51 with all solar bodies (including the four big asteroids ) for almost 3000 years and got this picture . Evolution of eccentricity and inclination are corresponding to the simuation above , but what a difference the Earth Moid makes after 400 yeras ! . Whereas the asteroid now stays at more than 0.5 AU it will be come closer than 0.01 AU to Earth ! . So the Kozai mechanism may induce additional threats !

Tony provided a while ago the following simulation http://www.orbitsimulator.com/gravity/articles/kozai.html showing a polar moon colliding into the Earth due to the Kozai Mechanism The plot herunder gives the evolution of the inclination vs. eccentricity of the simulation above ( The time goes from left to right ) . The plot was generated using the Gravsim output . One can see the moon performing small oscillations at low eccentricities initially These oscillations grow and eccentricity rises till at the end the moon collides with Earth . This evolution is not "straight forward " as at some times the moons eccentricity even recedes a liitle bit before it takes a jump forwards .